Abstract: This paper clarifies the fundamental principle of valid reasoning by dividing the principles of valid reasoning into deductions, abductions and inductions, and providing formalizations for each kind in terms of elementary probability theory and elementary set theory, both of which the paper presumes. It introduces two new probabilistic assumptions, one for abductions and one for inductions. If accepted, these assumptions make abductions and inductions into valid deductions depending on these assumptions in probability theory.
Sections:
(0) Introduction
(1) Deduction
(2) Abduction
(3) Induction
(4) Discussion of the abductive and inductive postulates
See also: Fundamental principles of invalid reasoning
Internet note: The fonts used are Verdana and StarMath.
"Probability is the guide to life".
Bishop Butler
(0) Introduction: In this paper I shall understand by reasoning any verbal inferencing using assumptions and conclusions, and by valid reasoning any instance of reasoning with a conclusion that is true if the assumptions given for it are true.
It is true  I shall assume  that there is more to human reasoning than verbal inferencing using assumptions and conclusions, and it is also true that reasoning may be very useful when it is not valid.
The reason to restrict myself to valid reasoning as defined is that most human reasoning can be cast in the form of verbal inferencing using assumptions and conclusions, and that valid reasoning has the very useful property of always leading from (presumptive) truths to (presumptive) truths.
This property guarantees that as long as one does use valid reasoning, one's conclusions are true whenever one's assumptions are true, and therefore also that one's assumptions are false if one's validly inferred conclusions are in fact false.
Thus valid reasoning gives us a way to find further truths from true premisses, and to find that assumed premisses are false if they validly imply conclusions that are false.
There are three basic kinds of reasoning, where reasoning involves argumentation of any kind using assumptions and inferences of conclusions:
1. Deductions: To find conclusions that follow from given assumptions
2. Abductions: To find assumptions from which given conclusions follow
3. Inductions: To confirm or infirm assumptions by showing their conclusions do (not) conform to the observable facts.
Normally in reasoning all three kinds are involved: We explain supposed facts by abductions; we check the abduced assumptions by deducing the facts they were to explain; and we test the assumptions arrived at inductively by deducing consequences and then revising the probabilities of the assumptions by probabilistic reasoning when these consequences are verified or falsified.
Here are some quotations of C.S. Peirce on the subject of abduction:
"Abduction. (..) "Hypothesis [or abduction] may be defined as an argument which proceeds upon the assumption that a character which is known necessarily to involve a certain number of others, may be probably predicated of any object which has all the characteristics which this character is known to involve." (5.276) "An abduction is [thus] a method of forming a general prediction." (2.269) But this prediction is always in reference to an observed fact; indeed, an abductive conclusion "is only justified by its explaining an observed fact." (1.89) If we enter a room containing a number of bags of beans and a table upon which there is a handful of white beans, and if, after some searching, we open a bag which contains white beans only, we may infer as a probability, or fair guess, that the handful was taken from this bag. This sort of inference is called making an hypothesis or abduction. (J. Feibleman, "An Introduction to the Philosophy of Charles S. Peirce", p. 1212. The numbers referred to are to paragraphs in Peirce's "Collected Papers".)
Now it is convenient to formalize the three types of inference I have stated in words above in the form of three general principles of inference. This can be done by using the theory of probability and some basic standard set theory, with both of which I assume familiarity in this paper. [1]
Since I assume familiarity with elementary probability theory, it should be remarked at the start that in this paper I have repressed all conventional "pr(.)"notation by relying on the fact that all probabilities can be written with a conditionalprobability sign when absolute probabilities are taken as probabilities conditional on the whole universe K. Thus, the probabilistic notation I use conforms to the following:
pr(sK&T) = sK&T and
pr(s) = pr(sK) = sK = s.
The reason to prefer the notation of the RHS (righthand side) of the equalities over that on their LHS is that using the format of the RHS avoids many redundant occurences of the functional operator "pr(.)". This device makes it possible to write elementary probabilitytheory almost exactly like propositional logic, where the formulas of probabilitytheory are distinguished by containing the mark of conditional probability and by being statements of (in)equalities.
It is important to see that the resulting probabilified version of propositional logic is considerably more subtle and has far more possibilities for different types of logical analysis than does standard propositional logic without probability. [2]
This last claim is supported by the following formalizations of the three fundamental principles of reasoning, that are given with the help of probabilitytheory. In each case I give a definition of the general conditions that makes the kind of reasoning valid in elementary probability theory and use this definition to give a rule of inference that conforms to the definition.
I start with deduction, because this is best known and simplest, and because I shall make abduction and induction into deductions that use special assumptions. Also I presume in the rest of this note the following notational conventions:
K = knowledge assumed, T = specific theory added to K, and s = specific statement.
K and T are supposed to be sets of statements, that can be rendered as a conjunction, and s a statement.
The main reasons to include K are that in fact in all reasoning (presumed) background knowledge is used, and that indeed often the (in)validity of one's reasoning is due to one's presumed background knowledge, and that this is very easily accomodated by the formalism of probability theory, namely as part of one's conditional probabilities.
(1) Deduction: To find conclusions that follow from given assumptions: sK=1 is a valid deduction from T given K =def
sK&T=1 & TK=1
Thus a valid deduction (using probability theory) is a probabilistically certain conclusion from a (presumptively) certain theory, assuming also background knowledge K. Of course, TK=1 may be withdrawn: All one often needs and can get is "if the theory is true, then given the knowledge we presume, it must also be true that ...". The general pattern of deductions is the familiar "If soandso is true, then suchandsuch is true, and soandso is true, therefore suchandsuch is a valid deduction."
And a deduction is the inference (where "A = B" = "B is a valid inference from premisses A") that conforms to:
sK&T=1 & TK=1 = sK=1
and is used in any case where one infers a conclusion from premisses (indeed also where in fact sK&T=1 is false, and therefore the deduction invalid, which we shall not consider in this paper).
In a valid deduction we infer a fact s as conclusion from a fact T, presuming K, and the fact s follows from K&T. The deductive reason for sK=1 is sK&T=1 & TK=1 i.e. s is true because it follows from K&T and K&T is true, presuming K.
There is nothing new here (if you know some probability theory and logic) apart from the facts that everything is stated in probabilistic terms and that the statement is explicit about the presence of some assumed background knowledge K. One advantage of including this reference to assumed background knowledge in a probabilistic setting is that if one has a valid deduction as defined, it is immediately obvious that if TK=x then sK&T=x, and thus one has a way of dealing explicitly with uncertainty of one's premisses. It also makes sense to point out that the assumption [(sK&T)=1] also allows for the possibility that s is a probabilistic statement, as in [((This die has a probability of 1/6 to fall with 3 facing)K&(This die is fair))=1]. [3]
It also is important to see that one has in fact very great liberties in deducing:
 one can choose whatever assumptions one pleases and
 one can assume whatever principle of inference one pleases
 provided only that one retains the criterion that a deduction of a conclusion from assumptions is valid precisely if in any case that the assumptions are true, so is the conclusion, and chooses one's principles of inference such that they satisfy this criterion.
An example of a deduction is: Let K be sufficient plane geometry to include Pythagoras' Theorem, and T the statements that X is a straightangled triangle in which the sides joined by the straigh angle have the lengths 3 and 4, and s be the statement that the side in X that joins the nonstraight angles has length 5.
And though one has very great liberties in deducing, part of the reason that this is so is that all deduction moves in the hypothetical realm of ifthen, with no deductve possibility of testing one's assumptions other than by validly deriving a known falsehood from them. It is this limitation that one seeks to escape by abduction and induction: Using only deductions and bivalent logic, one can only deduce conclusions, and make and refute assumptions, but one cannot infer assumptions nor support or undermine them.
(2) Abduction: The definition of abduction that I give relies on the definition of "TpsK" =def "T is positively relevant to s given K" that follows, together with two related definitions that will be useful below:
"TpsK" =def "T is positively relevant to s given K" =def "sK&T > sK&~T".
"TrsK" =def "T is relevant to s given K" =def "sK&T <> sK&~T"
"TisK" =def "T is irrelevant to s given K" =def "sK&T = sK&~T".
Now abduction is used to find assumptions from which a given conclusion follows, and so defined as follows using the probabilistic concept of (conditional) positive relevance:
TK&s=x is a valid abduction from s given K =def
TpsK & sK&T>=1/2 & sK=1 & TK=x
And so an abduction is an inference that conforms to:
sK&T>sK&~T & sK&T≥~sK&T & sK=1 & TK=x = TK&s=x
and is generally resorted to when one seeks to explain some puzzling fact s which one can not (one believes) derive by deducing it from one's presumed background knowledge. The present definition and rule of abduction claim that this is valid if in fact the theory T is relevant to and does support the truth of the puzzling fact s, and if in fact one has a probability for TK that may be low. Indeed, often abductions start as theories that would explain certain facts with a high probability if true, while the probability of the theory is not high.
In a valid abduction we infer a theory T as a possible explanation for a newly given fact s, presuming K, and we also infer that T has a certain probability, namely equal to the minimum probability of its proper consequences presuming K. The abductive reason for TK&s=x is that trsK & sK&TS1/2 & sK=1 & TK=x i.e. that it is true that T is a possible explanation for s precisely if it is true that T is positively relevant to s and that T makes s more probable than not and that s is true, all given K, while the least probable proper consequence of T given K has probability x. The general pattern of abductions is: "If theory T is positively relevant to s and the probability of s is at least 1/2 if T, and s is true and the probability of T is x, then T is a valid abduction for s, and T has probability x given s."
Given the definition of valid abduction and the premisses listed, it is an easy matter to verify that then it is deductively true, in probability theory, that T is a possible explanation for a newly given fact s presuming K. But to establish that TK=x we need some principle to settle the probability of a theory, since probability theory has no axioms sufficient to assign probabilities to theories, and we used the following:
The abductive postulate: TK=x IFF
x=q_{i}K&T & q_{i}Є{q: K&T=q & ~(K&~T=q) & (s)(K&T=s & ~(K&~T=s) > sK&T≥qK&T)}
This principle says that the probability of a theory T on presumed knowledge K equals x precisely if x is the minimum of T's proper consequences given K, where q is a proper consequence of K&T if it follows deductively from K&T but not from K&~T. (One needs proper consequences to avoid problems with just any statement with a low probability that happens to be true, for such a statement is a logical consequence of any theory.) It is a postulate because while probability theory implies that in the stated conditions TKRx it does not imply that TK=x. [4]
What is new here apart from stating everything probabilistically is that in fact two statements are inferred, of which one is deduced and one assumed. First it is deduced that it is true that T is a possible explanation of s, for which it is sufficient by the given definition that sK&T>sK&~T & sK&T≥~sK&T & sK=1. Next an assumption is made to settle the probability of a theory, namely that TK equals the minimum probability of its proper consequences, all presuming K. This is motivated by the facts that this minimum probability is the upper boundary of TK on probability theory; that probability theory by itself provides no consequences other than this to settle the probability of a theory; and that we need to assign a probability to theories to reason probabilistically with them. [5]
It should be noted that TK&s may be low, and that it still it may be a valid abduction, for this depends not on how low it is, but on T being positively relevant to s given K and making s given K at least 1/2. The possibility of 1/2 is included here to take care of whatever has that probability 1/2. Thus, that this coin fell heads half of the times it was thrown is explained by the valid abduction that this coin is fair (i.e. unloaded and with a heads and a tails side). But often abductive inference is used to find a theory to account for a surprising fact.
Also, if TK&s=x is small this suggests to do one (or more) of three things:
(1) Find another valid abduction for s with a higher probability or
(2) deduce a consequence of T and test T, in the hope to inductively confirm T and increase the probability of T or
(3) revise K so that the minimal proper consequence of T gets higher.
An example of an abduction is: Let K be elementary physics; let s the statement that a star S_{1} is not precisely where it should be if elementary physics is true; and let T be the theory that there is a hitherto unknown interstaller object S_{2} that effects where S_{1} precisely is.
The basic weaknesses of abductions are that (1) the only reason to infer their theoretical conclusions is that they are relevant to some puzzling facts and that (2) the theoretical conclusions inferred by abduction often have a low probability. This last weakness is addressed by induction.
(3) Induction: To confirm (or infirm) assumptions by showing their conclusions do (not) conform to the observable facts.
TK&s=TK*[sK&T:sK] is a valid induction from s given K =def
sK=1 & qK=1 & sK&q=sK & (q)(s)(TrsK > sK&q=sK IFF sK&T&q=sK&T)
And the induction is the inference
sK=1 & qK=1 &
(q)(s)(TrsK > sK&q=sK IFF sK&T&q=sK&T) =
TK&s=TK*[sK&T:sK]
although it may be clearer to write it like so, explicitly listing the probabilities that must be presumed:
sK=1 & sK&T=s_{1} & sK=s_{2} & TK=t_{1} & qK=1 & sK&q=sK &
(q)(s)(TrqK > sK&q=sK IFF sK&T&q=sK&T) =
TK&s=t_{1}*s_{1}:s_{2}
In a valid induction we infer that the probability of T given K&s equals TK*[sK&T:sK] from a true fact s that is relevant to T, while T satisfies the inductive postulate that any fact that is in fact irrelevant to anything T is relevant to remains irrelevant if T is true, all given K. [6]
What is new here is
The inductive postulate: (q)(s)(TrsK > (qK&s=qK IFF qK&T&s=qK&s))
This principle enables one to abstract from any fact that is not relevant when testing a theory. The way this works can be easily shown when we abstract for a moment from K. Then the principle turns into: [Trs > (qs=q IFF qT&s=qs)] and allows us to abstract from any q not relevant to any s that T is relevant to and that happens to be the case when T is tested, for now one can reason as follows: Ts&q = qT&s*T&s : s&q = qs*T&s : qs*s = T&s:s = Ts. Each step is deductively valid in probability theory, but the second step involves the inductive postulate to abstract from q. [7]
So in induction too there is an assumption involved, namely that the theory one uses has the property that any fact that is irrevelant in fact to any of the predictions of the theory is irrelevant also if the theory is true and conversely, all given K. This can be seen equivalently as the claim that the theory T is relevant in theory to everything it is relevant to in fact and nothing else. Indeed, the equivalence in the inductive postulate (qK&s=qK IFF qK&s&T=qK&s) can be conveniently read from left to right as: what is irrelevant in fact, also is irrelevant according to theory T, and from right to left as: what is irrelevant according to theory T is irrelevant in fact, all presuming K (i.e. sofar as we know).
It should be noted that TK&s may be any probability, including what T was before s came to be known, namely TK, if T is irrelevant to s given K. What matters for an inductive inference to be valid is the assumption that the inductive assumption is satisfied by T, for it is this that allows one to make an inductive inference about T given a new fact s.
An example of an induction is: Let K be standard medical science and s be the statement that test S is positive and T be the theory that you have cancer, and suppose sK&T=s_{1}=0.9 and sK&~T=s_{2}=0.3 and TK=t_{1}=0.01 then TK&s=t_{1}*s_{1}:s_{2}= 0.03. If the test is positive, your chance of having cancer is trebled, but if it was 1/100 to start with, then a positive result makes this 3/100. The inductive postulate enters because to credit the reasoning one must in fact assume that the test was properly done and that all manner of circumstances that occurred when it was done either are irrelevant to the theory or are accounted for by it. ("No madam, the lab assistant was not drunk, and no, it does not matter the test was done on a Tuesday in a leap year and no, your meditation exercises have nothing to do with outcome of the test." Etc.)
(4) Discussion of the abductive and inductive postulates:
The explanations I have given of abductive and inductive reasoning makes these valid deductions inside elementary probability theory, given certain postulates and definitions, that I also have given. In this section I want to consider briefly the postulates I made, and make a few remarks about the interesting epistemological status they share, namely of being practically necessary and corrigible.
A. The abductive postulate: The basic reasons to assume the abductive postulate are that one needs to arrive somehow at probabilities for theories, and that one has by probabilitytheory that TKRx, which the postulate strengthens to an identity provided x=q_{i}K&T & q_{i}Є{q: K&T=q & ~(K&~T=q) & (s)(K&T=s & ~(K&~T=s) > sK&T≥qK&T)}. For since whatever theories may denote (presumably: the infinite set of logical consequences of the assumptions of the theory), these cannot be counted in the same sense as e.g. occurences of "heads up" in sequences of throwns with a coin can be counted, and so there is no evident way to assign a probability to a theory.
The abductive postulate amounts to the assumptions that, first there is such a thing as the probability of a theory [8], and second that it may be initially and conveniently settled by supposing this probability equals the maximum of what it may be given in probability theory and such knowledge as one presumes, while one also fully expects that this initial probability will be adjusted by further reasoning and to be discovered new facts.
So the abductive postulate seems not so much a truth about nature as a truth about the ways and procedures human beings may use to discover the truth about nature. And indeed the abductive postulate seems safe and warranted in the sense that any probability it introduces can be  and usually will be  rationally corrected and adjusted, and that indeed it may be increased or decreased by inductions, and also by various other means indicated above.
An abductive postulate is needed because we need to have some factually based probability for theories that we decide are good explanations, if only to have a start to test them inductively.
B. The inductive postulate: The basic reason to assume the inductive postulate is that one needs some assumption to deal with the very many facts that are true besides the theory one is interested in testing, since each of these very many facts may be relevant to the truth of the fact one is interested in, and so that it seems a good demand to make of a theory to be true that it should truly and fully entail all it is relevant to.
Also, in fact this postulate seems to be necessarily true if human beings can come to know nature by testing such theories as they have, for all such tests must include knowledge of what is relevant to what is tested and in what degree it is relevant and also of what is irrelevant to it, for relevancies and irrelevancies are facts that are as real as the facts they concern. In brief: one just cannot rely on any experimental evidence if one cannot rely on one's abstraction from much of the surrounding factual details as irrelevant, which is necessary in any experiment. [9]
On the other hand, one cannot normally prove in complete or even considerable detail that any given theory that is to be tested in fact does correctly entail all that is relevant to it and does not entail as relevant anything that is in fact irrelevant. (Indeed, normally only a few known relevant factors are listed in any report of a scientific experiment together with an indication how these have been dealt with in the experimental setup. Yet any design of experiments must involve assumptions about factors that are relevant and that are irrelevant to what is to be tested.)
But since true theories must properly entail the true degrees of relevancies of their predictions, all one can do is to assume that one's theories do so, and to take care of all relevancies one does know.
So the inductive postulate seems not so much a truth about nature as a truth about the ways and procedures human beings use to discover the truth about nature, and one which is true to the extent human beings have true theories about nature, for true theories must satisfy the inductive postulate, even if no human being is able to survey all of the universe and establish all its presumed relevancies and irrelevancies are factually correct. And indeed the inductive postulate seems safe and warranted in the sense that any probabilities it introduces can be  and usually will be  rationally corrected and adjusted by later evidence. Also it suggests a reason for experiments that fail or turn out unexpected results: One may have disregarded as irrelevant some factor that is relevant i.e. one may have falsely assumed that one's theory T satisfied the inductive postulate. Finally, the inductive postulate is needed because in any experimental test of a theory we need to abstract from very many accompanying circumstances.
Summing up: It seems to me that both the abductive and inductive postulate are used tacitly in very much of human reasoning; it seems to me that both  or at least: assumptions much like them  are required by human beings if they want to arrive by reasoning at truths about nature; and it seems to me that both postulates have the interesting property that one must assume (something like) them in order to learn anything at all about the natural facts, while one is able to correct such inaccuracies as they may introduce, though this correction will take time and trouble, as does any scientific advance.
And the reason one must assume these principles is that in order to establish that one's theories are about nature (and are not merely fantasy), one must test one's theories, and to test them one needs both probabilities for theories and make as sure as one can that all that is relevant to what one tests is known, so that what one abstracts from as irrelevant to what one tests in theory indeed is irrelevant in fact.
Therefore it seems sensible to list these assumptions of scientific procedure explicitly, and to use them consciously wherever applicable. And finally it seems an interesting fact that there are, then, corrigible presumptive truths of procedure, that one needs to make in order to validly infer intermediate conclusions that are required on the way towards new natural knowledge.
Literature:
Ernest Adams: A Primer of Probability Theory
Arthur Burks: Chance, Cause and Reason
Paul Halmos: Naive Set Theory, and Measure Theory.
Charles Peirce: Collected Papers
G. Polya: Principles of Plausible Reasoning (2 volumes)
W.G. Wood & D.H. Martin: Experimental Method
Notes:
[1]: In fact I assume no more than standard elementary probability theory (summarizable as: What follows from Kolmogorov's axioms without any infinitary assumptions, or alternatively but equivalently: measure theory without infinitary assumptions) and standard elementary set theory. Good introductions to the former are by Adams and Burks, and a good introduction to set theory and to measure theory are by Halmos. Wood & Martin is a useful summary of principles involved in scientific experimental designs.
[2]: This is well explained by Adams and by Polya.
[3]: So while there is no need to formalize deductions inside probability theory it may be helpful to do so, especially when one wants to deal with uncertainties. Also, it is well to stress that the notation I use (and explained in the beginning) is useful, and more elegant and easier to read and use that the ordinary format using "pr(.)".
[4]: For in probability theory we have for any theory T and any statement s logically implied by T that the probability of T  assuming, as we do, that it exists and is consistent with probability theory  cannot be larger than the probability of s. In more or less standard notation: T = s > pr(T)Rpr(s).
[5]: These considerations do not logically imply the abductive postulate, and more can be said about the reasons to assume it, some of which is said in section (4).
[6]: That TK&s=TK*[sK&T:sK] in fact does follow by standard elementary probability theory and is well explained by Burks, with much more detail than is dealt with in the present paper. In any case, the basis of it all is the elementary theorem of probability theory to the effect that (abstracting from K for the moment) Ts > T IFF sT > s IFF T is positively relevant to s, while it also is a theorem that Ts = sT.T:s. So probability theory itself easily and elegantly enables inductive confirmation of a theory by its verified predictions, and falsification of a theory by its falsified predictation. But  though very interesting in principle  this is all presumed as known in the present paper. (See e.g. Burks and Polya.)
The inductive postulate is needed in any case s here is an empirical fact, that could be used as confirmation or infirmation, since such a fact s always wil be simultaneously true with very many other empirical facts, some of which may be relevant to it, and others which may not be relevant to it.
[7]: There is a lot more that can and should be said about the inductive postulate and its relations to the problems of induction as raised by Hume and Goodman, but the present paper is not the place for it, for it is not dedicated to the problems of induction, but to a clear and formal statement of the fundamental principles of valid reasoning.
[8]: It may be well to remark here that I do not presume the subjective or personal approach to probabilities, which makes probabilities depend on human betting quotients. These are arbitrary, and what I want, rather, is a probability based on the facts, in so far as these are (presumptively) known. These probabilities are not available for theories directly, since one cannot count the cases in which a theory is true and count the cases in which a theory is not true, and therefore I assume a theory has the probability of its least probable proper consequence, where this probability can be established experimentally by counting the cases pro and con.
[9]: In fact this is one side of the problem of induction that has been missed by many. One of the many was Karl Popper, whose whole philosophy of science founders on the fact that he did not see that to use experimental evidence requires an assumption about what is and is not relevant to it. Any experiment involves the assumption to the effect that whatever is happening at the time that is not considered by the theory that is tested is in fact irrelevant to whatever the theory implies. Without this assumption anything whatsoever that is happening may be quoted as "reason" that the experiment has the outcome it does, whatever the outcome is. And the assumption must be explicitly made because it may be mistaken and is involved in all testing of theories by their empirically checkable implications: Some of the factors that are not assumed to be relevant in fact may be relevant.
